## Properties of root subspaces

Let be a square matrix and let be its eigenvalue. As we know, the nonzero elements of the null space are the corresponding eigenvectors. This definition is generalized as follows.

**Definition 1**. The subspaces are called **root subspaces** of corresponding to

**Exercise 1**. a) Root subspaces are increasing:

(1) for all

and b) there is such that all inclusions (1) are strict for and

(2)

**Proof**. a) If for some then which shows that

b) (1) implies Since all root subspaces are contained in there are such that Let be the smallest such Then all inclusions (1) are strict for

Suppose for some Then there exists such that , that is, Put Then This means

that which contradicts the definition of

**Definition 2**. Property (2) can be called **stabilization**. The number from (2) is called a **height of the eigenvalue** .

**Exercise 2**. Let and let be the number from Exercise 1. Then

(3)

**Proof**. By the rank-nullity theorem applied to we have By Exercise 3, to prove (3) it is sufficient to establish that Let's assume that contains a nonzero vector Then we have for some We obtain two facts:

It follows that is a nonzero element of This contradicts (2). Hence, the assumption is wrong, and (3) follows.

**Exercise 3**. Both subspaces at the right of (3) are invariant with respect to

**Proof**. If then by commutativity of and we have so

Suppose so that for some Then

Exercise 3 means that, for the purpose of further analyzing we can consider its restrictions onto and

**Exercise 4**. The restriction of onto does not have eigenvalues other than

**Proof**. Suppose for some Since we have Then and . This implies

**Exercise 5**. The restriction of onto does not have as an eigenvalue (so that is invertible).

**Proof**. Suppose and Then for some and By Exercise 1 and This contradicts the choice of

## Leave a Reply

You must be logged in to post a comment.